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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    package org.apache.commons.math3.linear;<a name="line.17"></a>
<FONT color="green">018</FONT>    <a name="line.18"></a>
<FONT color="green">019</FONT>    import org.apache.commons.math3.exception.NumberIsTooLargeException;<a name="line.19"></a>
<FONT color="green">020</FONT>    import org.apache.commons.math3.exception.util.LocalizedFormats;<a name="line.20"></a>
<FONT color="green">021</FONT>    import org.apache.commons.math3.util.FastMath;<a name="line.21"></a>
<FONT color="green">022</FONT>    import org.apache.commons.math3.util.Precision;<a name="line.22"></a>
<FONT color="green">023</FONT>    <a name="line.23"></a>
<FONT color="green">024</FONT>    /**<a name="line.24"></a>
<FONT color="green">025</FONT>     * Calculates the compact Singular Value Decomposition of a matrix.<a name="line.25"></a>
<FONT color="green">026</FONT>     * &lt;p&gt;<a name="line.26"></a>
<FONT color="green">027</FONT>     * The Singular Value Decomposition of matrix A is a set of three matrices: U,<a name="line.27"></a>
<FONT color="green">028</FONT>     * &amp;Sigma; and V such that A = U &amp;times; &amp;Sigma; &amp;times; V&lt;sup&gt;T&lt;/sup&gt;. Let A be<a name="line.28"></a>
<FONT color="green">029</FONT>     * a m &amp;times; n matrix, then U is a m &amp;times; p orthogonal matrix, &amp;Sigma; is a<a name="line.29"></a>
<FONT color="green">030</FONT>     * p &amp;times; p diagonal matrix with positive or null elements, V is a p &amp;times;<a name="line.30"></a>
<FONT color="green">031</FONT>     * n orthogonal matrix (hence V&lt;sup&gt;T&lt;/sup&gt; is also orthogonal) where<a name="line.31"></a>
<FONT color="green">032</FONT>     * p=min(m,n).<a name="line.32"></a>
<FONT color="green">033</FONT>     * &lt;/p&gt;<a name="line.33"></a>
<FONT color="green">034</FONT>     * &lt;p&gt;This class is similar to the class with similar name from the<a name="line.34"></a>
<FONT color="green">035</FONT>     * &lt;a href="http://math.nist.gov/javanumerics/jama/"&gt;JAMA&lt;/a&gt; library, with the<a name="line.35"></a>
<FONT color="green">036</FONT>     * following changes:&lt;/p&gt;<a name="line.36"></a>
<FONT color="green">037</FONT>     * &lt;ul&gt;<a name="line.37"></a>
<FONT color="green">038</FONT>     *   &lt;li&gt;the {@code norm2} method which has been renamed as {@link #getNorm()<a name="line.38"></a>
<FONT color="green">039</FONT>     *   getNorm},&lt;/li&gt;<a name="line.39"></a>
<FONT color="green">040</FONT>     *   &lt;li&gt;the {@code cond} method which has been renamed as {@link<a name="line.40"></a>
<FONT color="green">041</FONT>     *   #getConditionNumber() getConditionNumber},&lt;/li&gt;<a name="line.41"></a>
<FONT color="green">042</FONT>     *   &lt;li&gt;the {@code rank} method which has been renamed as {@link #getRank()<a name="line.42"></a>
<FONT color="green">043</FONT>     *   getRank},&lt;/li&gt;<a name="line.43"></a>
<FONT color="green">044</FONT>     *   &lt;li&gt;a {@link #getUT() getUT} method has been added,&lt;/li&gt;<a name="line.44"></a>
<FONT color="green">045</FONT>     *   &lt;li&gt;a {@link #getVT() getVT} method has been added,&lt;/li&gt;<a name="line.45"></a>
<FONT color="green">046</FONT>     *   &lt;li&gt;a {@link #getSolver() getSolver} method has been added,&lt;/li&gt;<a name="line.46"></a>
<FONT color="green">047</FONT>     *   &lt;li&gt;a {@link #getCovariance(double) getCovariance} method has been added.&lt;/li&gt;<a name="line.47"></a>
<FONT color="green">048</FONT>     * &lt;/ul&gt;<a name="line.48"></a>
<FONT color="green">049</FONT>     * @see &lt;a href="http://mathworld.wolfram.com/SingularValueDecomposition.html"&gt;MathWorld&lt;/a&gt;<a name="line.49"></a>
<FONT color="green">050</FONT>     * @see &lt;a href="http://en.wikipedia.org/wiki/Singular_value_decomposition"&gt;Wikipedia&lt;/a&gt;<a name="line.50"></a>
<FONT color="green">051</FONT>     * @version $Id: SingularValueDecomposition.java 1416643 2012-12-03 19:37:14Z tn $<a name="line.51"></a>
<FONT color="green">052</FONT>     * @since 2.0 (changed to concrete class in 3.0)<a name="line.52"></a>
<FONT color="green">053</FONT>     */<a name="line.53"></a>
<FONT color="green">054</FONT>    public class SingularValueDecomposition {<a name="line.54"></a>
<FONT color="green">055</FONT>        /** Relative threshold for small singular values. */<a name="line.55"></a>
<FONT color="green">056</FONT>        private static final double EPS = 0x1.0p-52;<a name="line.56"></a>
<FONT color="green">057</FONT>        /** Absolute threshold for small singular values. */<a name="line.57"></a>
<FONT color="green">058</FONT>        private static final double TINY = 0x1.0p-966;<a name="line.58"></a>
<FONT color="green">059</FONT>        /** Computed singular values. */<a name="line.59"></a>
<FONT color="green">060</FONT>        private final double[] singularValues;<a name="line.60"></a>
<FONT color="green">061</FONT>        /** max(row dimension, column dimension). */<a name="line.61"></a>
<FONT color="green">062</FONT>        private final int m;<a name="line.62"></a>
<FONT color="green">063</FONT>        /** min(row dimension, column dimension). */<a name="line.63"></a>
<FONT color="green">064</FONT>        private final int n;<a name="line.64"></a>
<FONT color="green">065</FONT>        /** Indicator for transposed matrix. */<a name="line.65"></a>
<FONT color="green">066</FONT>        private final boolean transposed;<a name="line.66"></a>
<FONT color="green">067</FONT>        /** Cached value of U matrix. */<a name="line.67"></a>
<FONT color="green">068</FONT>        private final RealMatrix cachedU;<a name="line.68"></a>
<FONT color="green">069</FONT>        /** Cached value of transposed U matrix. */<a name="line.69"></a>
<FONT color="green">070</FONT>        private RealMatrix cachedUt;<a name="line.70"></a>
<FONT color="green">071</FONT>        /** Cached value of S (diagonal) matrix. */<a name="line.71"></a>
<FONT color="green">072</FONT>        private RealMatrix cachedS;<a name="line.72"></a>
<FONT color="green">073</FONT>        /** Cached value of V matrix. */<a name="line.73"></a>
<FONT color="green">074</FONT>        private final RealMatrix cachedV;<a name="line.74"></a>
<FONT color="green">075</FONT>        /** Cached value of transposed V matrix. */<a name="line.75"></a>
<FONT color="green">076</FONT>        private RealMatrix cachedVt;<a name="line.76"></a>
<FONT color="green">077</FONT>        /**<a name="line.77"></a>
<FONT color="green">078</FONT>         * Tolerance value for small singular values, calculated once we have<a name="line.78"></a>
<FONT color="green">079</FONT>         * populated "singularValues".<a name="line.79"></a>
<FONT color="green">080</FONT>         **/<a name="line.80"></a>
<FONT color="green">081</FONT>        private final double tol;<a name="line.81"></a>
<FONT color="green">082</FONT>    <a name="line.82"></a>
<FONT color="green">083</FONT>        /**<a name="line.83"></a>
<FONT color="green">084</FONT>         * Calculates the compact Singular Value Decomposition of the given matrix.<a name="line.84"></a>
<FONT color="green">085</FONT>         *<a name="line.85"></a>
<FONT color="green">086</FONT>         * @param matrix Matrix to decompose.<a name="line.86"></a>
<FONT color="green">087</FONT>         */<a name="line.87"></a>
<FONT color="green">088</FONT>        public SingularValueDecomposition(final RealMatrix matrix) {<a name="line.88"></a>
<FONT color="green">089</FONT>            final double[][] A;<a name="line.89"></a>
<FONT color="green">090</FONT>    <a name="line.90"></a>
<FONT color="green">091</FONT>             // "m" is always the largest dimension.<a name="line.91"></a>
<FONT color="green">092</FONT>            if (matrix.getRowDimension() &lt; matrix.getColumnDimension()) {<a name="line.92"></a>
<FONT color="green">093</FONT>                transposed = true;<a name="line.93"></a>
<FONT color="green">094</FONT>                A = matrix.transpose().getData();<a name="line.94"></a>
<FONT color="green">095</FONT>                m = matrix.getColumnDimension();<a name="line.95"></a>
<FONT color="green">096</FONT>                n = matrix.getRowDimension();<a name="line.96"></a>
<FONT color="green">097</FONT>            } else {<a name="line.97"></a>
<FONT color="green">098</FONT>                transposed = false;<a name="line.98"></a>
<FONT color="green">099</FONT>                A = matrix.getData();<a name="line.99"></a>
<FONT color="green">100</FONT>                m = matrix.getRowDimension();<a name="line.100"></a>
<FONT color="green">101</FONT>                n = matrix.getColumnDimension();<a name="line.101"></a>
<FONT color="green">102</FONT>            }<a name="line.102"></a>
<FONT color="green">103</FONT>    <a name="line.103"></a>
<FONT color="green">104</FONT>            singularValues = new double[n];<a name="line.104"></a>
<FONT color="green">105</FONT>            final double[][] U = new double[m][n];<a name="line.105"></a>
<FONT color="green">106</FONT>            final double[][] V = new double[n][n];<a name="line.106"></a>
<FONT color="green">107</FONT>            final double[] e = new double[n];<a name="line.107"></a>
<FONT color="green">108</FONT>            final double[] work = new double[m];<a name="line.108"></a>
<FONT color="green">109</FONT>            // Reduce A to bidiagonal form, storing the diagonal elements<a name="line.109"></a>
<FONT color="green">110</FONT>            // in s and the super-diagonal elements in e.<a name="line.110"></a>
<FONT color="green">111</FONT>            final int nct = FastMath.min(m - 1, n);<a name="line.111"></a>
<FONT color="green">112</FONT>            final int nrt = FastMath.max(0, n - 2);<a name="line.112"></a>
<FONT color="green">113</FONT>            for (int k = 0; k &lt; FastMath.max(nct, nrt); k++) {<a name="line.113"></a>
<FONT color="green">114</FONT>                if (k &lt; nct) {<a name="line.114"></a>
<FONT color="green">115</FONT>                    // Compute the transformation for the k-th column and<a name="line.115"></a>
<FONT color="green">116</FONT>                    // place the k-th diagonal in s[k].<a name="line.116"></a>
<FONT color="green">117</FONT>                    // Compute 2-norm of k-th column without under/overflow.<a name="line.117"></a>
<FONT color="green">118</FONT>                    singularValues[k] = 0;<a name="line.118"></a>
<FONT color="green">119</FONT>                    for (int i = k; i &lt; m; i++) {<a name="line.119"></a>
<FONT color="green">120</FONT>                        singularValues[k] = FastMath.hypot(singularValues[k], A[i][k]);<a name="line.120"></a>
<FONT color="green">121</FONT>                    }<a name="line.121"></a>
<FONT color="green">122</FONT>                    if (singularValues[k] != 0) {<a name="line.122"></a>
<FONT color="green">123</FONT>                        if (A[k][k] &lt; 0) {<a name="line.123"></a>
<FONT color="green">124</FONT>                            singularValues[k] = -singularValues[k];<a name="line.124"></a>
<FONT color="green">125</FONT>                        }<a name="line.125"></a>
<FONT color="green">126</FONT>                        for (int i = k; i &lt; m; i++) {<a name="line.126"></a>
<FONT color="green">127</FONT>                            A[i][k] /= singularValues[k];<a name="line.127"></a>
<FONT color="green">128</FONT>                        }<a name="line.128"></a>
<FONT color="green">129</FONT>                        A[k][k] += 1;<a name="line.129"></a>
<FONT color="green">130</FONT>                    }<a name="line.130"></a>
<FONT color="green">131</FONT>                    singularValues[k] = -singularValues[k];<a name="line.131"></a>
<FONT color="green">132</FONT>                }<a name="line.132"></a>
<FONT color="green">133</FONT>                for (int j = k + 1; j &lt; n; j++) {<a name="line.133"></a>
<FONT color="green">134</FONT>                    if (k &lt; nct &amp;&amp;<a name="line.134"></a>
<FONT color="green">135</FONT>                        singularValues[k] != 0) {<a name="line.135"></a>
<FONT color="green">136</FONT>                        // Apply the transformation.<a name="line.136"></a>
<FONT color="green">137</FONT>                        double t = 0;<a name="line.137"></a>
<FONT color="green">138</FONT>                        for (int i = k; i &lt; m; i++) {<a name="line.138"></a>
<FONT color="green">139</FONT>                            t += A[i][k] * A[i][j];<a name="line.139"></a>
<FONT color="green">140</FONT>                        }<a name="line.140"></a>
<FONT color="green">141</FONT>                        t = -t / A[k][k];<a name="line.141"></a>
<FONT color="green">142</FONT>                        for (int i = k; i &lt; m; i++) {<a name="line.142"></a>
<FONT color="green">143</FONT>                            A[i][j] += t * A[i][k];<a name="line.143"></a>
<FONT color="green">144</FONT>                        }<a name="line.144"></a>
<FONT color="green">145</FONT>                    }<a name="line.145"></a>
<FONT color="green">146</FONT>                    // Place the k-th row of A into e for the<a name="line.146"></a>
<FONT color="green">147</FONT>                    // subsequent calculation of the row transformation.<a name="line.147"></a>
<FONT color="green">148</FONT>                    e[j] = A[k][j];<a name="line.148"></a>
<FONT color="green">149</FONT>                }<a name="line.149"></a>
<FONT color="green">150</FONT>                if (k &lt; nct) {<a name="line.150"></a>
<FONT color="green">151</FONT>                    // Place the transformation in U for subsequent back<a name="line.151"></a>
<FONT color="green">152</FONT>                    // multiplication.<a name="line.152"></a>
<FONT color="green">153</FONT>                    for (int i = k; i &lt; m; i++) {<a name="line.153"></a>
<FONT color="green">154</FONT>                        U[i][k] = A[i][k];<a name="line.154"></a>
<FONT color="green">155</FONT>                    }<a name="line.155"></a>
<FONT color="green">156</FONT>                }<a name="line.156"></a>
<FONT color="green">157</FONT>                if (k &lt; nrt) {<a name="line.157"></a>
<FONT color="green">158</FONT>                    // Compute the k-th row transformation and place the<a name="line.158"></a>
<FONT color="green">159</FONT>                    // k-th super-diagonal in e[k].<a name="line.159"></a>
<FONT color="green">160</FONT>                    // Compute 2-norm without under/overflow.<a name="line.160"></a>
<FONT color="green">161</FONT>                    e[k] = 0;<a name="line.161"></a>
<FONT color="green">162</FONT>                    for (int i = k + 1; i &lt; n; i++) {<a name="line.162"></a>
<FONT color="green">163</FONT>                        e[k] = FastMath.hypot(e[k], e[i]);<a name="line.163"></a>
<FONT color="green">164</FONT>                    }<a name="line.164"></a>
<FONT color="green">165</FONT>                    if (e[k] != 0) {<a name="line.165"></a>
<FONT color="green">166</FONT>                        if (e[k + 1] &lt; 0) {<a name="line.166"></a>
<FONT color="green">167</FONT>                            e[k] = -e[k];<a name="line.167"></a>
<FONT color="green">168</FONT>                        }<a name="line.168"></a>
<FONT color="green">169</FONT>                        for (int i = k + 1; i &lt; n; i++) {<a name="line.169"></a>
<FONT color="green">170</FONT>                            e[i] /= e[k];<a name="line.170"></a>
<FONT color="green">171</FONT>                        }<a name="line.171"></a>
<FONT color="green">172</FONT>                        e[k + 1] += 1;<a name="line.172"></a>
<FONT color="green">173</FONT>                    }<a name="line.173"></a>
<FONT color="green">174</FONT>                    e[k] = -e[k];<a name="line.174"></a>
<FONT color="green">175</FONT>                    if (k + 1 &lt; m &amp;&amp;<a name="line.175"></a>
<FONT color="green">176</FONT>                        e[k] != 0) {<a name="line.176"></a>
<FONT color="green">177</FONT>                        // Apply the transformation.<a name="line.177"></a>
<FONT color="green">178</FONT>                        for (int i = k + 1; i &lt; m; i++) {<a name="line.178"></a>
<FONT color="green">179</FONT>                            work[i] = 0;<a name="line.179"></a>
<FONT color="green">180</FONT>                        }<a name="line.180"></a>
<FONT color="green">181</FONT>                        for (int j = k + 1; j &lt; n; j++) {<a name="line.181"></a>
<FONT color="green">182</FONT>                            for (int i = k + 1; i &lt; m; i++) {<a name="line.182"></a>
<FONT color="green">183</FONT>                                work[i] += e[j] * A[i][j];<a name="line.183"></a>
<FONT color="green">184</FONT>                            }<a name="line.184"></a>
<FONT color="green">185</FONT>                        }<a name="line.185"></a>
<FONT color="green">186</FONT>                        for (int j = k + 1; j &lt; n; j++) {<a name="line.186"></a>
<FONT color="green">187</FONT>                            final double t = -e[j] / e[k + 1];<a name="line.187"></a>
<FONT color="green">188</FONT>                            for (int i = k + 1; i &lt; m; i++) {<a name="line.188"></a>
<FONT color="green">189</FONT>                                A[i][j] += t * work[i];<a name="line.189"></a>
<FONT color="green">190</FONT>                            }<a name="line.190"></a>
<FONT color="green">191</FONT>                        }<a name="line.191"></a>
<FONT color="green">192</FONT>                    }<a name="line.192"></a>
<FONT color="green">193</FONT>    <a name="line.193"></a>
<FONT color="green">194</FONT>                    // Place the transformation in V for subsequent<a name="line.194"></a>
<FONT color="green">195</FONT>                    // back multiplication.<a name="line.195"></a>
<FONT color="green">196</FONT>                    for (int i = k + 1; i &lt; n; i++) {<a name="line.196"></a>
<FONT color="green">197</FONT>                        V[i][k] = e[i];<a name="line.197"></a>
<FONT color="green">198</FONT>                    }<a name="line.198"></a>
<FONT color="green">199</FONT>                }<a name="line.199"></a>
<FONT color="green">200</FONT>            }<a name="line.200"></a>
<FONT color="green">201</FONT>            // Set up the final bidiagonal matrix or order p.<a name="line.201"></a>
<FONT color="green">202</FONT>            int p = n;<a name="line.202"></a>
<FONT color="green">203</FONT>            if (nct &lt; n) {<a name="line.203"></a>
<FONT color="green">204</FONT>                singularValues[nct] = A[nct][nct];<a name="line.204"></a>
<FONT color="green">205</FONT>            }<a name="line.205"></a>
<FONT color="green">206</FONT>            if (m &lt; p) {<a name="line.206"></a>
<FONT color="green">207</FONT>                singularValues[p - 1] = 0;<a name="line.207"></a>
<FONT color="green">208</FONT>            }<a name="line.208"></a>
<FONT color="green">209</FONT>            if (nrt + 1 &lt; p) {<a name="line.209"></a>
<FONT color="green">210</FONT>                e[nrt] = A[nrt][p - 1];<a name="line.210"></a>
<FONT color="green">211</FONT>            }<a name="line.211"></a>
<FONT color="green">212</FONT>            e[p - 1] = 0;<a name="line.212"></a>
<FONT color="green">213</FONT>    <a name="line.213"></a>
<FONT color="green">214</FONT>            // Generate U.<a name="line.214"></a>
<FONT color="green">215</FONT>            for (int j = nct; j &lt; n; j++) {<a name="line.215"></a>
<FONT color="green">216</FONT>                for (int i = 0; i &lt; m; i++) {<a name="line.216"></a>
<FONT color="green">217</FONT>                    U[i][j] = 0;<a name="line.217"></a>
<FONT color="green">218</FONT>                }<a name="line.218"></a>
<FONT color="green">219</FONT>                U[j][j] = 1;<a name="line.219"></a>
<FONT color="green">220</FONT>            }<a name="line.220"></a>
<FONT color="green">221</FONT>            for (int k = nct - 1; k &gt;= 0; k--) {<a name="line.221"></a>
<FONT color="green">222</FONT>                if (singularValues[k] != 0) {<a name="line.222"></a>
<FONT color="green">223</FONT>                    for (int j = k + 1; j &lt; n; j++) {<a name="line.223"></a>
<FONT color="green">224</FONT>                        double t = 0;<a name="line.224"></a>
<FONT color="green">225</FONT>                        for (int i = k; i &lt; m; i++) {<a name="line.225"></a>
<FONT color="green">226</FONT>                            t += U[i][k] * U[i][j];<a name="line.226"></a>
<FONT color="green">227</FONT>                        }<a name="line.227"></a>
<FONT color="green">228</FONT>                        t = -t / U[k][k];<a name="line.228"></a>
<FONT color="green">229</FONT>                        for (int i = k; i &lt; m; i++) {<a name="line.229"></a>
<FONT color="green">230</FONT>                            U[i][j] += t * U[i][k];<a name="line.230"></a>
<FONT color="green">231</FONT>                        }<a name="line.231"></a>
<FONT color="green">232</FONT>                    }<a name="line.232"></a>
<FONT color="green">233</FONT>                    for (int i = k; i &lt; m; i++) {<a name="line.233"></a>
<FONT color="green">234</FONT>                        U[i][k] = -U[i][k];<a name="line.234"></a>
<FONT color="green">235</FONT>                    }<a name="line.235"></a>
<FONT color="green">236</FONT>                    U[k][k] = 1 + U[k][k];<a name="line.236"></a>
<FONT color="green">237</FONT>                    for (int i = 0; i &lt; k - 1; i++) {<a name="line.237"></a>
<FONT color="green">238</FONT>                        U[i][k] = 0;<a name="line.238"></a>
<FONT color="green">239</FONT>                    }<a name="line.239"></a>
<FONT color="green">240</FONT>                } else {<a name="line.240"></a>
<FONT color="green">241</FONT>                    for (int i = 0; i &lt; m; i++) {<a name="line.241"></a>
<FONT color="green">242</FONT>                        U[i][k] = 0;<a name="line.242"></a>
<FONT color="green">243</FONT>                    }<a name="line.243"></a>
<FONT color="green">244</FONT>                    U[k][k] = 1;<a name="line.244"></a>
<FONT color="green">245</FONT>                }<a name="line.245"></a>
<FONT color="green">246</FONT>            }<a name="line.246"></a>
<FONT color="green">247</FONT>    <a name="line.247"></a>
<FONT color="green">248</FONT>            // Generate V.<a name="line.248"></a>
<FONT color="green">249</FONT>            for (int k = n - 1; k &gt;= 0; k--) {<a name="line.249"></a>
<FONT color="green">250</FONT>                if (k &lt; nrt &amp;&amp;<a name="line.250"></a>
<FONT color="green">251</FONT>                    e[k] != 0) {<a name="line.251"></a>
<FONT color="green">252</FONT>                    for (int j = k + 1; j &lt; n; j++) {<a name="line.252"></a>
<FONT color="green">253</FONT>                        double t = 0;<a name="line.253"></a>
<FONT color="green">254</FONT>                        for (int i = k + 1; i &lt; n; i++) {<a name="line.254"></a>
<FONT color="green">255</FONT>                            t += V[i][k] * V[i][j];<a name="line.255"></a>
<FONT color="green">256</FONT>                        }<a name="line.256"></a>
<FONT color="green">257</FONT>                        t = -t / V[k + 1][k];<a name="line.257"></a>
<FONT color="green">258</FONT>                        for (int i = k + 1; i &lt; n; i++) {<a name="line.258"></a>
<FONT color="green">259</FONT>                            V[i][j] += t * V[i][k];<a name="line.259"></a>
<FONT color="green">260</FONT>                        }<a name="line.260"></a>
<FONT color="green">261</FONT>                    }<a name="line.261"></a>
<FONT color="green">262</FONT>                }<a name="line.262"></a>
<FONT color="green">263</FONT>                for (int i = 0; i &lt; n; i++) {<a name="line.263"></a>
<FONT color="green">264</FONT>                    V[i][k] = 0;<a name="line.264"></a>
<FONT color="green">265</FONT>                }<a name="line.265"></a>
<FONT color="green">266</FONT>                V[k][k] = 1;<a name="line.266"></a>
<FONT color="green">267</FONT>            }<a name="line.267"></a>
<FONT color="green">268</FONT>    <a name="line.268"></a>
<FONT color="green">269</FONT>            // Main iteration loop for the singular values.<a name="line.269"></a>
<FONT color="green">270</FONT>            final int pp = p - 1;<a name="line.270"></a>
<FONT color="green">271</FONT>            int iter = 0;<a name="line.271"></a>
<FONT color="green">272</FONT>            while (p &gt; 0) {<a name="line.272"></a>
<FONT color="green">273</FONT>                int k;<a name="line.273"></a>
<FONT color="green">274</FONT>                int kase;<a name="line.274"></a>
<FONT color="green">275</FONT>                // Here is where a test for too many iterations would go.<a name="line.275"></a>
<FONT color="green">276</FONT>                // This section of the program inspects for<a name="line.276"></a>
<FONT color="green">277</FONT>                // negligible elements in the s and e arrays.  On<a name="line.277"></a>
<FONT color="green">278</FONT>                // completion the variables kase and k are set as follows.<a name="line.278"></a>
<FONT color="green">279</FONT>                // kase = 1     if s(p) and e[k-1] are negligible and k&lt;p<a name="line.279"></a>
<FONT color="green">280</FONT>                // kase = 2     if s(k) is negligible and k&lt;p<a name="line.280"></a>
<FONT color="green">281</FONT>                // kase = 3     if e[k-1] is negligible, k&lt;p, and<a name="line.281"></a>
<FONT color="green">282</FONT>                //              s(k), ..., s(p) are not negligible (qr step).<a name="line.282"></a>
<FONT color="green">283</FONT>                // kase = 4     if e(p-1) is negligible (convergence).<a name="line.283"></a>
<FONT color="green">284</FONT>                for (k = p - 2; k &gt;= 0; k--) {<a name="line.284"></a>
<FONT color="green">285</FONT>                    final double threshold<a name="line.285"></a>
<FONT color="green">286</FONT>                        = TINY + EPS * (FastMath.abs(singularValues[k]) +<a name="line.286"></a>
<FONT color="green">287</FONT>                                        FastMath.abs(singularValues[k + 1]));<a name="line.287"></a>
<FONT color="green">288</FONT>                    if (FastMath.abs(e[k]) &lt;= threshold) {<a name="line.288"></a>
<FONT color="green">289</FONT>                        e[k] = 0;<a name="line.289"></a>
<FONT color="green">290</FONT>                        break;<a name="line.290"></a>
<FONT color="green">291</FONT>                    }<a name="line.291"></a>
<FONT color="green">292</FONT>                }<a name="line.292"></a>
<FONT color="green">293</FONT>    <a name="line.293"></a>
<FONT color="green">294</FONT>                if (k == p - 2) {<a name="line.294"></a>
<FONT color="green">295</FONT>                    kase = 4;<a name="line.295"></a>
<FONT color="green">296</FONT>                } else {<a name="line.296"></a>
<FONT color="green">297</FONT>                    int ks;<a name="line.297"></a>
<FONT color="green">298</FONT>                    for (ks = p - 1; ks &gt;= k; ks--) {<a name="line.298"></a>
<FONT color="green">299</FONT>                        if (ks == k) {<a name="line.299"></a>
<FONT color="green">300</FONT>                            break;<a name="line.300"></a>
<FONT color="green">301</FONT>                        }<a name="line.301"></a>
<FONT color="green">302</FONT>                        final double t = (ks != p ? FastMath.abs(e[ks]) : 0) +<a name="line.302"></a>
<FONT color="green">303</FONT>                            (ks != k + 1 ? FastMath.abs(e[ks - 1]) : 0);<a name="line.303"></a>
<FONT color="green">304</FONT>                        if (FastMath.abs(singularValues[ks]) &lt;= TINY + EPS * t) {<a name="line.304"></a>
<FONT color="green">305</FONT>                            singularValues[ks] = 0;<a name="line.305"></a>
<FONT color="green">306</FONT>                            break;<a name="line.306"></a>
<FONT color="green">307</FONT>                        }<a name="line.307"></a>
<FONT color="green">308</FONT>                    }<a name="line.308"></a>
<FONT color="green">309</FONT>                    if (ks == k) {<a name="line.309"></a>
<FONT color="green">310</FONT>                        kase = 3;<a name="line.310"></a>
<FONT color="green">311</FONT>                    } else if (ks == p - 1) {<a name="line.311"></a>
<FONT color="green">312</FONT>                        kase = 1;<a name="line.312"></a>
<FONT color="green">313</FONT>                    } else {<a name="line.313"></a>
<FONT color="green">314</FONT>                        kase = 2;<a name="line.314"></a>
<FONT color="green">315</FONT>                        k = ks;<a name="line.315"></a>
<FONT color="green">316</FONT>                    }<a name="line.316"></a>
<FONT color="green">317</FONT>                }<a name="line.317"></a>
<FONT color="green">318</FONT>                k++;<a name="line.318"></a>
<FONT color="green">319</FONT>                // Perform the task indicated by kase.<a name="line.319"></a>
<FONT color="green">320</FONT>                switch (kase) {<a name="line.320"></a>
<FONT color="green">321</FONT>                    // Deflate negligible s(p).<a name="line.321"></a>
<FONT color="green">322</FONT>                    case 1: {<a name="line.322"></a>
<FONT color="green">323</FONT>                        double f = e[p - 2];<a name="line.323"></a>
<FONT color="green">324</FONT>                        e[p - 2] = 0;<a name="line.324"></a>
<FONT color="green">325</FONT>                        for (int j = p - 2; j &gt;= k; j--) {<a name="line.325"></a>
<FONT color="green">326</FONT>                            double t = FastMath.hypot(singularValues[j], f);<a name="line.326"></a>
<FONT color="green">327</FONT>                            final double cs = singularValues[j] / t;<a name="line.327"></a>
<FONT color="green">328</FONT>                            final double sn = f / t;<a name="line.328"></a>
<FONT color="green">329</FONT>                            singularValues[j] = t;<a name="line.329"></a>
<FONT color="green">330</FONT>                            if (j != k) {<a name="line.330"></a>
<FONT color="green">331</FONT>                                f = -sn * e[j - 1];<a name="line.331"></a>
<FONT color="green">332</FONT>                                e[j - 1] = cs * e[j - 1];<a name="line.332"></a>
<FONT color="green">333</FONT>                            }<a name="line.333"></a>
<FONT color="green">334</FONT>    <a name="line.334"></a>
<FONT color="green">335</FONT>                            for (int i = 0; i &lt; n; i++) {<a name="line.335"></a>
<FONT color="green">336</FONT>                                t = cs * V[i][j] + sn * V[i][p - 1];<a name="line.336"></a>
<FONT color="green">337</FONT>                                V[i][p - 1] = -sn * V[i][j] + cs * V[i][p - 1];<a name="line.337"></a>
<FONT color="green">338</FONT>                                V[i][j] = t;<a name="line.338"></a>
<FONT color="green">339</FONT>                            }<a name="line.339"></a>
<FONT color="green">340</FONT>                        }<a name="line.340"></a>
<FONT color="green">341</FONT>                    }<a name="line.341"></a>
<FONT color="green">342</FONT>                    break;<a name="line.342"></a>
<FONT color="green">343</FONT>                    // Split at negligible s(k).<a name="line.343"></a>
<FONT color="green">344</FONT>                    case 2: {<a name="line.344"></a>
<FONT color="green">345</FONT>                        double f = e[k - 1];<a name="line.345"></a>
<FONT color="green">346</FONT>                        e[k - 1] = 0;<a name="line.346"></a>
<FONT color="green">347</FONT>                        for (int j = k; j &lt; p; j++) {<a name="line.347"></a>
<FONT color="green">348</FONT>                            double t = FastMath.hypot(singularValues[j], f);<a name="line.348"></a>
<FONT color="green">349</FONT>                            final double cs = singularValues[j] / t;<a name="line.349"></a>
<FONT color="green">350</FONT>                            final double sn = f / t;<a name="line.350"></a>
<FONT color="green">351</FONT>                            singularValues[j] = t;<a name="line.351"></a>
<FONT color="green">352</FONT>                            f = -sn * e[j];<a name="line.352"></a>
<FONT color="green">353</FONT>                            e[j] = cs * e[j];<a name="line.353"></a>
<FONT color="green">354</FONT>    <a name="line.354"></a>
<FONT color="green">355</FONT>                            for (int i = 0; i &lt; m; i++) {<a name="line.355"></a>
<FONT color="green">356</FONT>                                t = cs * U[i][j] + sn * U[i][k - 1];<a name="line.356"></a>
<FONT color="green">357</FONT>                                U[i][k - 1] = -sn * U[i][j] + cs * U[i][k - 1];<a name="line.357"></a>
<FONT color="green">358</FONT>                                U[i][j] = t;<a name="line.358"></a>
<FONT color="green">359</FONT>                            }<a name="line.359"></a>
<FONT color="green">360</FONT>                        }<a name="line.360"></a>
<FONT color="green">361</FONT>                    }<a name="line.361"></a>
<FONT color="green">362</FONT>                    break;<a name="line.362"></a>
<FONT color="green">363</FONT>                    // Perform one qr step.<a name="line.363"></a>
<FONT color="green">364</FONT>                    case 3: {<a name="line.364"></a>
<FONT color="green">365</FONT>                        // Calculate the shift.<a name="line.365"></a>
<FONT color="green">366</FONT>                        final double maxPm1Pm2 = FastMath.max(FastMath.abs(singularValues[p - 1]),<a name="line.366"></a>
<FONT color="green">367</FONT>                                                              FastMath.abs(singularValues[p - 2]));<a name="line.367"></a>
<FONT color="green">368</FONT>                        final double scale = FastMath.max(FastMath.max(FastMath.max(maxPm1Pm2,<a name="line.368"></a>
<FONT color="green">369</FONT>                                                                                    FastMath.abs(e[p - 2])),<a name="line.369"></a>
<FONT color="green">370</FONT>                                                                       FastMath.abs(singularValues[k])),<a name="line.370"></a>
<FONT color="green">371</FONT>                                                          FastMath.abs(e[k]));<a name="line.371"></a>
<FONT color="green">372</FONT>                        final double sp = singularValues[p - 1] / scale;<a name="line.372"></a>
<FONT color="green">373</FONT>                        final double spm1 = singularValues[p - 2] / scale;<a name="line.373"></a>
<FONT color="green">374</FONT>                        final double epm1 = e[p - 2] / scale;<a name="line.374"></a>
<FONT color="green">375</FONT>                        final double sk = singularValues[k] / scale;<a name="line.375"></a>
<FONT color="green">376</FONT>                        final double ek = e[k] / scale;<a name="line.376"></a>
<FONT color="green">377</FONT>                        final double b = ((spm1 + sp) * (spm1 - sp) + epm1 * epm1) / 2.0;<a name="line.377"></a>
<FONT color="green">378</FONT>                        final double c = (sp * epm1) * (sp * epm1);<a name="line.378"></a>
<FONT color="green">379</FONT>                        double shift = 0;<a name="line.379"></a>
<FONT color="green">380</FONT>                        if (b != 0 ||<a name="line.380"></a>
<FONT color="green">381</FONT>                            c != 0) {<a name="line.381"></a>
<FONT color="green">382</FONT>                            shift = FastMath.sqrt(b * b + c);<a name="line.382"></a>
<FONT color="green">383</FONT>                            if (b &lt; 0) {<a name="line.383"></a>
<FONT color="green">384</FONT>                                shift = -shift;<a name="line.384"></a>
<FONT color="green">385</FONT>                            }<a name="line.385"></a>
<FONT color="green">386</FONT>                            shift = c / (b + shift);<a name="line.386"></a>
<FONT color="green">387</FONT>                        }<a name="line.387"></a>
<FONT color="green">388</FONT>                        double f = (sk + sp) * (sk - sp) + shift;<a name="line.388"></a>
<FONT color="green">389</FONT>                        double g = sk * ek;<a name="line.389"></a>
<FONT color="green">390</FONT>                        // Chase zeros.<a name="line.390"></a>
<FONT color="green">391</FONT>                        for (int j = k; j &lt; p - 1; j++) {<a name="line.391"></a>
<FONT color="green">392</FONT>                            double t = FastMath.hypot(f, g);<a name="line.392"></a>
<FONT color="green">393</FONT>                            double cs = f / t;<a name="line.393"></a>
<FONT color="green">394</FONT>                            double sn = g / t;<a name="line.394"></a>
<FONT color="green">395</FONT>                            if (j != k) {<a name="line.395"></a>
<FONT color="green">396</FONT>                                e[j - 1] = t;<a name="line.396"></a>
<FONT color="green">397</FONT>                            }<a name="line.397"></a>
<FONT color="green">398</FONT>                            f = cs * singularValues[j] + sn * e[j];<a name="line.398"></a>
<FONT color="green">399</FONT>                            e[j] = cs * e[j] - sn * singularValues[j];<a name="line.399"></a>
<FONT color="green">400</FONT>                            g = sn * singularValues[j + 1];<a name="line.400"></a>
<FONT color="green">401</FONT>                            singularValues[j + 1] = cs * singularValues[j + 1];<a name="line.401"></a>
<FONT color="green">402</FONT>    <a name="line.402"></a>
<FONT color="green">403</FONT>                            for (int i = 0; i &lt; n; i++) {<a name="line.403"></a>
<FONT color="green">404</FONT>                                t = cs * V[i][j] + sn * V[i][j + 1];<a name="line.404"></a>
<FONT color="green">405</FONT>                                V[i][j + 1] = -sn * V[i][j] + cs * V[i][j + 1];<a name="line.405"></a>
<FONT color="green">406</FONT>                                V[i][j] = t;<a name="line.406"></a>
<FONT color="green">407</FONT>                            }<a name="line.407"></a>
<FONT color="green">408</FONT>                            t = FastMath.hypot(f, g);<a name="line.408"></a>
<FONT color="green">409</FONT>                            cs = f / t;<a name="line.409"></a>
<FONT color="green">410</FONT>                            sn = g / t;<a name="line.410"></a>
<FONT color="green">411</FONT>                            singularValues[j] = t;<a name="line.411"></a>
<FONT color="green">412</FONT>                            f = cs * e[j] + sn * singularValues[j + 1];<a name="line.412"></a>
<FONT color="green">413</FONT>                            singularValues[j + 1] = -sn * e[j] + cs * singularValues[j + 1];<a name="line.413"></a>
<FONT color="green">414</FONT>                            g = sn * e[j + 1];<a name="line.414"></a>
<FONT color="green">415</FONT>                            e[j + 1] = cs * e[j + 1];<a name="line.415"></a>
<FONT color="green">416</FONT>                            if (j &lt; m - 1) {<a name="line.416"></a>
<FONT color="green">417</FONT>                                for (int i = 0; i &lt; m; i++) {<a name="line.417"></a>
<FONT color="green">418</FONT>                                    t = cs * U[i][j] + sn * U[i][j + 1];<a name="line.418"></a>
<FONT color="green">419</FONT>                                    U[i][j + 1] = -sn * U[i][j] + cs * U[i][j + 1];<a name="line.419"></a>
<FONT color="green">420</FONT>                                    U[i][j] = t;<a name="line.420"></a>
<FONT color="green">421</FONT>                                }<a name="line.421"></a>
<FONT color="green">422</FONT>                            }<a name="line.422"></a>
<FONT color="green">423</FONT>                        }<a name="line.423"></a>
<FONT color="green">424</FONT>                        e[p - 2] = f;<a name="line.424"></a>
<FONT color="green">425</FONT>                        iter = iter + 1;<a name="line.425"></a>
<FONT color="green">426</FONT>                    }<a name="line.426"></a>
<FONT color="green">427</FONT>                    break;<a name="line.427"></a>
<FONT color="green">428</FONT>                    // Convergence.<a name="line.428"></a>
<FONT color="green">429</FONT>                    default: {<a name="line.429"></a>
<FONT color="green">430</FONT>                        // Make the singular values positive.<a name="line.430"></a>
<FONT color="green">431</FONT>                        if (singularValues[k] &lt;= 0) {<a name="line.431"></a>
<FONT color="green">432</FONT>                            singularValues[k] = singularValues[k] &lt; 0 ? -singularValues[k] : 0;<a name="line.432"></a>
<FONT color="green">433</FONT>    <a name="line.433"></a>
<FONT color="green">434</FONT>                            for (int i = 0; i &lt;= pp; i++) {<a name="line.434"></a>
<FONT color="green">435</FONT>                                V[i][k] = -V[i][k];<a name="line.435"></a>
<FONT color="green">436</FONT>                            }<a name="line.436"></a>
<FONT color="green">437</FONT>                        }<a name="line.437"></a>
<FONT color="green">438</FONT>                        // Order the singular values.<a name="line.438"></a>
<FONT color="green">439</FONT>                        while (k &lt; pp) {<a name="line.439"></a>
<FONT color="green">440</FONT>                            if (singularValues[k] &gt;= singularValues[k + 1]) {<a name="line.440"></a>
<FONT color="green">441</FONT>                                break;<a name="line.441"></a>
<FONT color="green">442</FONT>                            }<a name="line.442"></a>
<FONT color="green">443</FONT>                            double t = singularValues[k];<a name="line.443"></a>
<FONT color="green">444</FONT>                            singularValues[k] = singularValues[k + 1];<a name="line.444"></a>
<FONT color="green">445</FONT>                            singularValues[k + 1] = t;<a name="line.445"></a>
<FONT color="green">446</FONT>                            if (k &lt; n - 1) {<a name="line.446"></a>
<FONT color="green">447</FONT>                                for (int i = 0; i &lt; n; i++) {<a name="line.447"></a>
<FONT color="green">448</FONT>                                    t = V[i][k + 1];<a name="line.448"></a>
<FONT color="green">449</FONT>                                    V[i][k + 1] = V[i][k];<a name="line.449"></a>
<FONT color="green">450</FONT>                                    V[i][k] = t;<a name="line.450"></a>
<FONT color="green">451</FONT>                                }<a name="line.451"></a>
<FONT color="green">452</FONT>                            }<a name="line.452"></a>
<FONT color="green">453</FONT>                            if (k &lt; m - 1) {<a name="line.453"></a>
<FONT color="green">454</FONT>                                for (int i = 0; i &lt; m; i++) {<a name="line.454"></a>
<FONT color="green">455</FONT>                                    t = U[i][k + 1];<a name="line.455"></a>
<FONT color="green">456</FONT>                                    U[i][k + 1] = U[i][k];<a name="line.456"></a>
<FONT color="green">457</FONT>                                    U[i][k] = t;<a name="line.457"></a>
<FONT color="green">458</FONT>                                }<a name="line.458"></a>
<FONT color="green">459</FONT>                            }<a name="line.459"></a>
<FONT color="green">460</FONT>                            k++;<a name="line.460"></a>
<FONT color="green">461</FONT>                        }<a name="line.461"></a>
<FONT color="green">462</FONT>                        iter = 0;<a name="line.462"></a>
<FONT color="green">463</FONT>                        p--;<a name="line.463"></a>
<FONT color="green">464</FONT>                    }<a name="line.464"></a>
<FONT color="green">465</FONT>                    break;<a name="line.465"></a>
<FONT color="green">466</FONT>                }<a name="line.466"></a>
<FONT color="green">467</FONT>            }<a name="line.467"></a>
<FONT color="green">468</FONT>    <a name="line.468"></a>
<FONT color="green">469</FONT>            // Set the small value tolerance used to calculate rank and pseudo-inverse<a name="line.469"></a>
<FONT color="green">470</FONT>            tol = FastMath.max(m * singularValues[0] * EPS,<a name="line.470"></a>
<FONT color="green">471</FONT>                               FastMath.sqrt(Precision.SAFE_MIN));<a name="line.471"></a>
<FONT color="green">472</FONT>    <a name="line.472"></a>
<FONT color="green">473</FONT>            if (!transposed) {<a name="line.473"></a>
<FONT color="green">474</FONT>                cachedU = MatrixUtils.createRealMatrix(U);<a name="line.474"></a>
<FONT color="green">475</FONT>                cachedV = MatrixUtils.createRealMatrix(V);<a name="line.475"></a>
<FONT color="green">476</FONT>            } else {<a name="line.476"></a>
<FONT color="green">477</FONT>                cachedU = MatrixUtils.createRealMatrix(V);<a name="line.477"></a>
<FONT color="green">478</FONT>                cachedV = MatrixUtils.createRealMatrix(U);<a name="line.478"></a>
<FONT color="green">479</FONT>            }<a name="line.479"></a>
<FONT color="green">480</FONT>        }<a name="line.480"></a>
<FONT color="green">481</FONT>    <a name="line.481"></a>
<FONT color="green">482</FONT>        /**<a name="line.482"></a>
<FONT color="green">483</FONT>         * Returns the matrix U of the decomposition.<a name="line.483"></a>
<FONT color="green">484</FONT>         * &lt;p&gt;U is an orthogonal matrix, i.e. its transpose is also its inverse.&lt;/p&gt;<a name="line.484"></a>
<FONT color="green">485</FONT>         * @return the U matrix<a name="line.485"></a>
<FONT color="green">486</FONT>         * @see #getUT()<a name="line.486"></a>
<FONT color="green">487</FONT>         */<a name="line.487"></a>
<FONT color="green">488</FONT>        public RealMatrix getU() {<a name="line.488"></a>
<FONT color="green">489</FONT>            // return the cached matrix<a name="line.489"></a>
<FONT color="green">490</FONT>            return cachedU;<a name="line.490"></a>
<FONT color="green">491</FONT>    <a name="line.491"></a>
<FONT color="green">492</FONT>        }<a name="line.492"></a>
<FONT color="green">493</FONT>    <a name="line.493"></a>
<FONT color="green">494</FONT>        /**<a name="line.494"></a>
<FONT color="green">495</FONT>         * Returns the transpose of the matrix U of the decomposition.<a name="line.495"></a>
<FONT color="green">496</FONT>         * &lt;p&gt;U is an orthogonal matrix, i.e. its transpose is also its inverse.&lt;/p&gt;<a name="line.496"></a>
<FONT color="green">497</FONT>         * @return the U matrix (or null if decomposed matrix is singular)<a name="line.497"></a>
<FONT color="green">498</FONT>         * @see #getU()<a name="line.498"></a>
<FONT color="green">499</FONT>         */<a name="line.499"></a>
<FONT color="green">500</FONT>        public RealMatrix getUT() {<a name="line.500"></a>
<FONT color="green">501</FONT>            if (cachedUt == null) {<a name="line.501"></a>
<FONT color="green">502</FONT>                cachedUt = getU().transpose();<a name="line.502"></a>
<FONT color="green">503</FONT>            }<a name="line.503"></a>
<FONT color="green">504</FONT>            // return the cached matrix<a name="line.504"></a>
<FONT color="green">505</FONT>            return cachedUt;<a name="line.505"></a>
<FONT color="green">506</FONT>        }<a name="line.506"></a>
<FONT color="green">507</FONT>    <a name="line.507"></a>
<FONT color="green">508</FONT>        /**<a name="line.508"></a>
<FONT color="green">509</FONT>         * Returns the diagonal matrix &amp;Sigma; of the decomposition.<a name="line.509"></a>
<FONT color="green">510</FONT>         * &lt;p&gt;&amp;Sigma; is a diagonal matrix. The singular values are provided in<a name="line.510"></a>
<FONT color="green">511</FONT>         * non-increasing order, for compatibility with Jama.&lt;/p&gt;<a name="line.511"></a>
<FONT color="green">512</FONT>         * @return the &amp;Sigma; matrix<a name="line.512"></a>
<FONT color="green">513</FONT>         */<a name="line.513"></a>
<FONT color="green">514</FONT>        public RealMatrix getS() {<a name="line.514"></a>
<FONT color="green">515</FONT>            if (cachedS == null) {<a name="line.515"></a>
<FONT color="green">516</FONT>                // cache the matrix for subsequent calls<a name="line.516"></a>
<FONT color="green">517</FONT>                cachedS = MatrixUtils.createRealDiagonalMatrix(singularValues);<a name="line.517"></a>
<FONT color="green">518</FONT>            }<a name="line.518"></a>
<FONT color="green">519</FONT>            return cachedS;<a name="line.519"></a>
<FONT color="green">520</FONT>        }<a name="line.520"></a>
<FONT color="green">521</FONT>    <a name="line.521"></a>
<FONT color="green">522</FONT>        /**<a name="line.522"></a>
<FONT color="green">523</FONT>         * Returns the diagonal elements of the matrix &amp;Sigma; of the decomposition.<a name="line.523"></a>
<FONT color="green">524</FONT>         * &lt;p&gt;The singular values are provided in non-increasing order, for<a name="line.524"></a>
<FONT color="green">525</FONT>         * compatibility with Jama.&lt;/p&gt;<a name="line.525"></a>
<FONT color="green">526</FONT>         * @return the diagonal elements of the &amp;Sigma; matrix<a name="line.526"></a>
<FONT color="green">527</FONT>         */<a name="line.527"></a>
<FONT color="green">528</FONT>        public double[] getSingularValues() {<a name="line.528"></a>
<FONT color="green">529</FONT>            return singularValues.clone();<a name="line.529"></a>
<FONT color="green">530</FONT>        }<a name="line.530"></a>
<FONT color="green">531</FONT>    <a name="line.531"></a>
<FONT color="green">532</FONT>        /**<a name="line.532"></a>
<FONT color="green">533</FONT>         * Returns the matrix V of the decomposition.<a name="line.533"></a>
<FONT color="green">534</FONT>         * &lt;p&gt;V is an orthogonal matrix, i.e. its transpose is also its inverse.&lt;/p&gt;<a name="line.534"></a>
<FONT color="green">535</FONT>         * @return the V matrix (or null if decomposed matrix is singular)<a name="line.535"></a>
<FONT color="green">536</FONT>         * @see #getVT()<a name="line.536"></a>
<FONT color="green">537</FONT>         */<a name="line.537"></a>
<FONT color="green">538</FONT>        public RealMatrix getV() {<a name="line.538"></a>
<FONT color="green">539</FONT>            // return the cached matrix<a name="line.539"></a>
<FONT color="green">540</FONT>            return cachedV;<a name="line.540"></a>
<FONT color="green">541</FONT>        }<a name="line.541"></a>
<FONT color="green">542</FONT>    <a name="line.542"></a>
<FONT color="green">543</FONT>        /**<a name="line.543"></a>
<FONT color="green">544</FONT>         * Returns the transpose of the matrix V of the decomposition.<a name="line.544"></a>
<FONT color="green">545</FONT>         * &lt;p&gt;V is an orthogonal matrix, i.e. its transpose is also its inverse.&lt;/p&gt;<a name="line.545"></a>
<FONT color="green">546</FONT>         * @return the V matrix (or null if decomposed matrix is singular)<a name="line.546"></a>
<FONT color="green">547</FONT>         * @see #getV()<a name="line.547"></a>
<FONT color="green">548</FONT>         */<a name="line.548"></a>
<FONT color="green">549</FONT>        public RealMatrix getVT() {<a name="line.549"></a>
<FONT color="green">550</FONT>            if (cachedVt == null) {<a name="line.550"></a>
<FONT color="green">551</FONT>                cachedVt = getV().transpose();<a name="line.551"></a>
<FONT color="green">552</FONT>            }<a name="line.552"></a>
<FONT color="green">553</FONT>            // return the cached matrix<a name="line.553"></a>
<FONT color="green">554</FONT>            return cachedVt;<a name="line.554"></a>
<FONT color="green">555</FONT>        }<a name="line.555"></a>
<FONT color="green">556</FONT>    <a name="line.556"></a>
<FONT color="green">557</FONT>        /**<a name="line.557"></a>
<FONT color="green">558</FONT>         * Returns the n &amp;times; n covariance matrix.<a name="line.558"></a>
<FONT color="green">559</FONT>         * &lt;p&gt;The covariance matrix is V &amp;times; J &amp;times; V&lt;sup&gt;T&lt;/sup&gt;<a name="line.559"></a>
<FONT color="green">560</FONT>         * where J is the diagonal matrix of the inverse of the squares of<a name="line.560"></a>
<FONT color="green">561</FONT>         * the singular values.&lt;/p&gt;<a name="line.561"></a>
<FONT color="green">562</FONT>         * @param minSingularValue value below which singular values are ignored<a name="line.562"></a>
<FONT color="green">563</FONT>         * (a 0 or negative value implies all singular value will be used)<a name="line.563"></a>
<FONT color="green">564</FONT>         * @return covariance matrix<a name="line.564"></a>
<FONT color="green">565</FONT>         * @exception IllegalArgumentException if minSingularValue is larger than<a name="line.565"></a>
<FONT color="green">566</FONT>         * the largest singular value, meaning all singular values are ignored<a name="line.566"></a>
<FONT color="green">567</FONT>         */<a name="line.567"></a>
<FONT color="green">568</FONT>        public RealMatrix getCovariance(final double minSingularValue) {<a name="line.568"></a>
<FONT color="green">569</FONT>            // get the number of singular values to consider<a name="line.569"></a>
<FONT color="green">570</FONT>            final int p = singularValues.length;<a name="line.570"></a>
<FONT color="green">571</FONT>            int dimension = 0;<a name="line.571"></a>
<FONT color="green">572</FONT>            while (dimension &lt; p &amp;&amp;<a name="line.572"></a>
<FONT color="green">573</FONT>                   singularValues[dimension] &gt;= minSingularValue) {<a name="line.573"></a>
<FONT color="green">574</FONT>                ++dimension;<a name="line.574"></a>
<FONT color="green">575</FONT>            }<a name="line.575"></a>
<FONT color="green">576</FONT>    <a name="line.576"></a>
<FONT color="green">577</FONT>            if (dimension == 0) {<a name="line.577"></a>
<FONT color="green">578</FONT>                throw new NumberIsTooLargeException(LocalizedFormats.TOO_LARGE_CUTOFF_SINGULAR_VALUE,<a name="line.578"></a>
<FONT color="green">579</FONT>                                                    minSingularValue, singularValues[0], true);<a name="line.579"></a>
<FONT color="green">580</FONT>            }<a name="line.580"></a>
<FONT color="green">581</FONT>    <a name="line.581"></a>
<FONT color="green">582</FONT>            final double[][] data = new double[dimension][p];<a name="line.582"></a>
<FONT color="green">583</FONT>            getVT().walkInOptimizedOrder(new DefaultRealMatrixPreservingVisitor() {<a name="line.583"></a>
<FONT color="green">584</FONT>                /** {@inheritDoc} */<a name="line.584"></a>
<FONT color="green">585</FONT>                @Override<a name="line.585"></a>
<FONT color="green">586</FONT>                public void visit(final int row, final int column,<a name="line.586"></a>
<FONT color="green">587</FONT>                        final double value) {<a name="line.587"></a>
<FONT color="green">588</FONT>                    data[row][column] = value / singularValues[row];<a name="line.588"></a>
<FONT color="green">589</FONT>                }<a name="line.589"></a>
<FONT color="green">590</FONT>            }, 0, dimension - 1, 0, p - 1);<a name="line.590"></a>
<FONT color="green">591</FONT>    <a name="line.591"></a>
<FONT color="green">592</FONT>            RealMatrix jv = new Array2DRowRealMatrix(data, false);<a name="line.592"></a>
<FONT color="green">593</FONT>            return jv.transpose().multiply(jv);<a name="line.593"></a>
<FONT color="green">594</FONT>        }<a name="line.594"></a>
<FONT color="green">595</FONT>    <a name="line.595"></a>
<FONT color="green">596</FONT>        /**<a name="line.596"></a>
<FONT color="green">597</FONT>         * Returns the L&lt;sub&gt;2&lt;/sub&gt; norm of the matrix.<a name="line.597"></a>
<FONT color="green">598</FONT>         * &lt;p&gt;The L&lt;sub&gt;2&lt;/sub&gt; norm is max(|A &amp;times; u|&lt;sub&gt;2&lt;/sub&gt; /<a name="line.598"></a>
<FONT color="green">599</FONT>         * |u|&lt;sub&gt;2&lt;/sub&gt;), where |.|&lt;sub&gt;2&lt;/sub&gt; denotes the vectorial 2-norm<a name="line.599"></a>
<FONT color="green">600</FONT>         * (i.e. the traditional euclidian norm).&lt;/p&gt;<a name="line.600"></a>
<FONT color="green">601</FONT>         * @return norm<a name="line.601"></a>
<FONT color="green">602</FONT>         */<a name="line.602"></a>
<FONT color="green">603</FONT>        public double getNorm() {<a name="line.603"></a>
<FONT color="green">604</FONT>            return singularValues[0];<a name="line.604"></a>
<FONT color="green">605</FONT>        }<a name="line.605"></a>
<FONT color="green">606</FONT>    <a name="line.606"></a>
<FONT color="green">607</FONT>        /**<a name="line.607"></a>
<FONT color="green">608</FONT>         * Return the condition number of the matrix.<a name="line.608"></a>
<FONT color="green">609</FONT>         * @return condition number of the matrix<a name="line.609"></a>
<FONT color="green">610</FONT>         */<a name="line.610"></a>
<FONT color="green">611</FONT>        public double getConditionNumber() {<a name="line.611"></a>
<FONT color="green">612</FONT>            return singularValues[0] / singularValues[n - 1];<a name="line.612"></a>
<FONT color="green">613</FONT>        }<a name="line.613"></a>
<FONT color="green">614</FONT>    <a name="line.614"></a>
<FONT color="green">615</FONT>        /**<a name="line.615"></a>
<FONT color="green">616</FONT>         * Computes the inverse of the condition number.<a name="line.616"></a>
<FONT color="green">617</FONT>         * In cases of rank deficiency, the {@link #getConditionNumber() condition<a name="line.617"></a>
<FONT color="green">618</FONT>         * number} will become undefined.<a name="line.618"></a>
<FONT color="green">619</FONT>         *<a name="line.619"></a>
<FONT color="green">620</FONT>         * @return the inverse of the condition number.<a name="line.620"></a>
<FONT color="green">621</FONT>         */<a name="line.621"></a>
<FONT color="green">622</FONT>        public double getInverseConditionNumber() {<a name="line.622"></a>
<FONT color="green">623</FONT>            return singularValues[n - 1] / singularValues[0];<a name="line.623"></a>
<FONT color="green">624</FONT>        }<a name="line.624"></a>
<FONT color="green">625</FONT>    <a name="line.625"></a>
<FONT color="green">626</FONT>        /**<a name="line.626"></a>
<FONT color="green">627</FONT>         * Return the effective numerical matrix rank.<a name="line.627"></a>
<FONT color="green">628</FONT>         * &lt;p&gt;The effective numerical rank is the number of non-negligible<a name="line.628"></a>
<FONT color="green">629</FONT>         * singular values. The threshold used to identify non-negligible<a name="line.629"></a>
<FONT color="green">630</FONT>         * terms is max(m,n) &amp;times; ulp(s&lt;sub&gt;1&lt;/sub&gt;) where ulp(s&lt;sub&gt;1&lt;/sub&gt;)<a name="line.630"></a>
<FONT color="green">631</FONT>         * is the least significant bit of the largest singular value.&lt;/p&gt;<a name="line.631"></a>
<FONT color="green">632</FONT>         * @return effective numerical matrix rank<a name="line.632"></a>
<FONT color="green">633</FONT>         */<a name="line.633"></a>
<FONT color="green">634</FONT>        public int getRank() {<a name="line.634"></a>
<FONT color="green">635</FONT>            int r = 0;<a name="line.635"></a>
<FONT color="green">636</FONT>            for (int i = 0; i &lt; singularValues.length; i++) {<a name="line.636"></a>
<FONT color="green">637</FONT>                if (singularValues[i] &gt; tol) {<a name="line.637"></a>
<FONT color="green">638</FONT>                    r++;<a name="line.638"></a>
<FONT color="green">639</FONT>                }<a name="line.639"></a>
<FONT color="green">640</FONT>            }<a name="line.640"></a>
<FONT color="green">641</FONT>            return r;<a name="line.641"></a>
<FONT color="green">642</FONT>        }<a name="line.642"></a>
<FONT color="green">643</FONT>    <a name="line.643"></a>
<FONT color="green">644</FONT>        /**<a name="line.644"></a>
<FONT color="green">645</FONT>         * Get a solver for finding the A &amp;times; X = B solution in least square sense.<a name="line.645"></a>
<FONT color="green">646</FONT>         * @return a solver<a name="line.646"></a>
<FONT color="green">647</FONT>         */<a name="line.647"></a>
<FONT color="green">648</FONT>        public DecompositionSolver getSolver() {<a name="line.648"></a>
<FONT color="green">649</FONT>            return new Solver(singularValues, getUT(), getV(), getRank() == m, tol);<a name="line.649"></a>
<FONT color="green">650</FONT>        }<a name="line.650"></a>
<FONT color="green">651</FONT>    <a name="line.651"></a>
<FONT color="green">652</FONT>        /** Specialized solver. */<a name="line.652"></a>
<FONT color="green">653</FONT>        private static class Solver implements DecompositionSolver {<a name="line.653"></a>
<FONT color="green">654</FONT>            /** Pseudo-inverse of the initial matrix. */<a name="line.654"></a>
<FONT color="green">655</FONT>            private final RealMatrix pseudoInverse;<a name="line.655"></a>
<FONT color="green">656</FONT>            /** Singularity indicator. */<a name="line.656"></a>
<FONT color="green">657</FONT>            private boolean nonSingular;<a name="line.657"></a>
<FONT color="green">658</FONT>    <a name="line.658"></a>
<FONT color="green">659</FONT>            /**<a name="line.659"></a>
<FONT color="green">660</FONT>             * Build a solver from decomposed matrix.<a name="line.660"></a>
<FONT color="green">661</FONT>             *<a name="line.661"></a>
<FONT color="green">662</FONT>             * @param singularValues Singular values.<a name="line.662"></a>
<FONT color="green">663</FONT>             * @param uT U&lt;sup&gt;T&lt;/sup&gt; matrix of the decomposition.<a name="line.663"></a>
<FONT color="green">664</FONT>             * @param v V matrix of the decomposition.<a name="line.664"></a>
<FONT color="green">665</FONT>             * @param nonSingular Singularity indicator.<a name="line.665"></a>
<FONT color="green">666</FONT>             * @param tol tolerance for singular values<a name="line.666"></a>
<FONT color="green">667</FONT>             */<a name="line.667"></a>
<FONT color="green">668</FONT>            private Solver(final double[] singularValues, final RealMatrix uT,<a name="line.668"></a>
<FONT color="green">669</FONT>                           final RealMatrix v, final boolean nonSingular, final double tol) {<a name="line.669"></a>
<FONT color="green">670</FONT>                final double[][] suT = uT.getData();<a name="line.670"></a>
<FONT color="green">671</FONT>                for (int i = 0; i &lt; singularValues.length; ++i) {<a name="line.671"></a>
<FONT color="green">672</FONT>                    final double a;<a name="line.672"></a>
<FONT color="green">673</FONT>                    if (singularValues[i] &gt; tol) {<a name="line.673"></a>
<FONT color="green">674</FONT>                        a = 1 / singularValues[i];<a name="line.674"></a>
<FONT color="green">675</FONT>                    } else {<a name="line.675"></a>
<FONT color="green">676</FONT>                        a = 0;<a name="line.676"></a>
<FONT color="green">677</FONT>                    }<a name="line.677"></a>
<FONT color="green">678</FONT>                    final double[] suTi = suT[i];<a name="line.678"></a>
<FONT color="green">679</FONT>                    for (int j = 0; j &lt; suTi.length; ++j) {<a name="line.679"></a>
<FONT color="green">680</FONT>                        suTi[j] *= a;<a name="line.680"></a>
<FONT color="green">681</FONT>                    }<a name="line.681"></a>
<FONT color="green">682</FONT>                }<a name="line.682"></a>
<FONT color="green">683</FONT>                pseudoInverse = v.multiply(new Array2DRowRealMatrix(suT, false));<a name="line.683"></a>
<FONT color="green">684</FONT>                this.nonSingular = nonSingular;<a name="line.684"></a>
<FONT color="green">685</FONT>            }<a name="line.685"></a>
<FONT color="green">686</FONT>    <a name="line.686"></a>
<FONT color="green">687</FONT>            /**<a name="line.687"></a>
<FONT color="green">688</FONT>             * Solve the linear equation A &amp;times; X = B in least square sense.<a name="line.688"></a>
<FONT color="green">689</FONT>             * &lt;p&gt;<a name="line.689"></a>
<FONT color="green">690</FONT>             * The m&amp;times;n matrix A may not be square, the solution X is such that<a name="line.690"></a>
<FONT color="green">691</FONT>             * ||A &amp;times; X - B|| is minimal.<a name="line.691"></a>
<FONT color="green">692</FONT>             * &lt;/p&gt;<a name="line.692"></a>
<FONT color="green">693</FONT>             * @param b Right-hand side of the equation A &amp;times; X = B<a name="line.693"></a>
<FONT color="green">694</FONT>             * @return a vector X that minimizes the two norm of A &amp;times; X - B<a name="line.694"></a>
<FONT color="green">695</FONT>             * @throws org.apache.commons.math3.exception.DimensionMismatchException<a name="line.695"></a>
<FONT color="green">696</FONT>             * if the matrices dimensions do not match.<a name="line.696"></a>
<FONT color="green">697</FONT>             */<a name="line.697"></a>
<FONT color="green">698</FONT>            public RealVector solve(final RealVector b) {<a name="line.698"></a>
<FONT color="green">699</FONT>                return pseudoInverse.operate(b);<a name="line.699"></a>
<FONT color="green">700</FONT>            }<a name="line.700"></a>
<FONT color="green">701</FONT>    <a name="line.701"></a>
<FONT color="green">702</FONT>            /**<a name="line.702"></a>
<FONT color="green">703</FONT>             * Solve the linear equation A &amp;times; X = B in least square sense.<a name="line.703"></a>
<FONT color="green">704</FONT>             * &lt;p&gt;<a name="line.704"></a>
<FONT color="green">705</FONT>             * The m&amp;times;n matrix A may not be square, the solution X is such that<a name="line.705"></a>
<FONT color="green">706</FONT>             * ||A &amp;times; X - B|| is minimal.<a name="line.706"></a>
<FONT color="green">707</FONT>             * &lt;/p&gt;<a name="line.707"></a>
<FONT color="green">708</FONT>             *<a name="line.708"></a>
<FONT color="green">709</FONT>             * @param b Right-hand side of the equation A &amp;times; X = B<a name="line.709"></a>
<FONT color="green">710</FONT>             * @return a matrix X that minimizes the two norm of A &amp;times; X - B<a name="line.710"></a>
<FONT color="green">711</FONT>             * @throws org.apache.commons.math3.exception.DimensionMismatchException<a name="line.711"></a>
<FONT color="green">712</FONT>             * if the matrices dimensions do not match.<a name="line.712"></a>
<FONT color="green">713</FONT>             */<a name="line.713"></a>
<FONT color="green">714</FONT>            public RealMatrix solve(final RealMatrix b) {<a name="line.714"></a>
<FONT color="green">715</FONT>                return pseudoInverse.multiply(b);<a name="line.715"></a>
<FONT color="green">716</FONT>            }<a name="line.716"></a>
<FONT color="green">717</FONT>    <a name="line.717"></a>
<FONT color="green">718</FONT>            /**<a name="line.718"></a>
<FONT color="green">719</FONT>             * Check if the decomposed matrix is non-singular.<a name="line.719"></a>
<FONT color="green">720</FONT>             *<a name="line.720"></a>
<FONT color="green">721</FONT>             * @return {@code true} if the decomposed matrix is non-singular.<a name="line.721"></a>
<FONT color="green">722</FONT>             */<a name="line.722"></a>
<FONT color="green">723</FONT>            public boolean isNonSingular() {<a name="line.723"></a>
<FONT color="green">724</FONT>                return nonSingular;<a name="line.724"></a>
<FONT color="green">725</FONT>            }<a name="line.725"></a>
<FONT color="green">726</FONT>    <a name="line.726"></a>
<FONT color="green">727</FONT>            /**<a name="line.727"></a>
<FONT color="green">728</FONT>             * Get the pseudo-inverse of the decomposed matrix.<a name="line.728"></a>
<FONT color="green">729</FONT>             *<a name="line.729"></a>
<FONT color="green">730</FONT>             * @return the inverse matrix.<a name="line.730"></a>
<FONT color="green">731</FONT>             */<a name="line.731"></a>
<FONT color="green">732</FONT>            public RealMatrix getInverse() {<a name="line.732"></a>
<FONT color="green">733</FONT>                return pseudoInverse;<a name="line.733"></a>
<FONT color="green">734</FONT>            }<a name="line.734"></a>
<FONT color="green">735</FONT>        }<a name="line.735"></a>
<FONT color="green">736</FONT>    }<a name="line.736"></a>




























































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